The behaviour of square functions from ergodic theory in $L^{\infty}$

TL;DR

This paper investigates the behavior of ergodic theory square functions in $L^{ abla}$, revealing unboundedness on certain functions, non-membership in BMO space, and establishing boundedness from $L^{ abla}_c$ to BMO, solving an open problem.

Contribution

It demonstrates unboundedness of square functions on some $L^{ abla}$ functions, constructs examples outside BMO, and proves uniform boundedness in $L^p$, addressing an open question.

Findings

Existence of $f$ with $Sf=\infty$ on a set of positive measure.

Existence of compactly supported $f$ with $S_\mathcal{I}f$ not in BMO.

Boundedness of $S$ from $L^{\infty}_c$ to BMO.

Abstract

In this paper, we analyze carefully the behaviour in $L^\infty(\mathbb R)$ of the square functions $S$ and $S_\mathcal I$'s, originating from ergodic theory. Firstly, we show that we can find some function $f\in L^\infty(\mathbb{R})$, such that $Sf$ equals infinity on a nonzero measure set. Secondly, we can find compact supported function $f\in L^\infty(\mathbb{R})$ and $\mathcal I$ such that $S_\mathcal{I} f$ does not belong to $BMO$ space. Finally, we show that $S$ is bounded from $L^{\infty}_c$ to $BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl in \cite{JKRW98}. That is, $S_\mathcal I$ are uniformly bounded in $L^p(\mathbb R)$ with respect to $\mathcal I$ for $2<p<\infty$.